Meanders and Dyck-Path Billiards

We study a statistic $\mathsf{traj}$ on the ordered pairs $(P,Q)$ of Dyck paths of size $n$, which counts the number of billiard trajectories in the grid polygon enclosed by $P$ and $-Q$, where $-Q$ is the path obtained by reflecting $Q$ over the ground line. It turns out to coincide with the component statistic of meanders. In terms of grid polygon, we establish an involution on the set of such ordered pairs $(P,Q)$ which either increases or decreases $\mathsf{traj}(P,Q)$ by 1. This proves a result by Di Francesco--Golinelli--Guitter that the numbers of semimeanders (meanders, respectively) of order $n$ with even and odd numbers of components are equal if $n$ is even and differ by a Catalan number (the square of a Catalan number, respectively) if $n$ is odd. Some results about $(-1)$-evaluation of the generating functions for the statistic $\mathsf{traj}$ on restricted sets of Dyck paths are also presented.